On the degree of the inverse of quadratic permutation polynomial interleavers

An integral component of a turbo code is a carefully designed interleaver. Interleavers based on quadratic permutation polynomials (modulo N) were introduced by Sun and Takeshita. They have several good properties and were selected to be used in a future cellular phone system. Ryu and Takeshita later initiated the study of the related deinterleavers. Here we extend this latter work and introduce a very efficient method for computing the (degree of the) lowest degree polynomial giving the deinterleaver. Our approach is based on combining two techniques. The Chinese remainder theorem allows us to study one prime factor of N at a time. Our other technique is to first present the inverse function as a power series, and then turn that power series to a low degree polynomial using a Gröbner basis of the ideal of polynomials vanishing modulo a prime power.